Honestly, I only suggested the mozzarella as a way to block wetter items from soaking the crust. Almost any cheese will do. Grated smoked gouda would probably be great.
Wet pizza is a pet peeve of mine. I really hate being in a group situation and saying I am vegetarian and then some other people deciding to share my pizza and thusly try to order with/for me. It's one thing to think, ok, we should eat some vegetables for lunch, it's quite another to pile raw tomatoes, bell peppers, mushrooms, onions, basil, etc. on top of a pizza. -All you get is soup with a wan crust that resembles a paper-thin noodle.
Anyway, back to breakfast pizza.... I am not advocating a lot of mild cheese per se, I just want to make sure there is some kind of moisture barrier to keep the eggs and perhaps other things from soaking the crust. Also, beware of too much cheese. I have made the mistake in the past (starting with raw dough, not pre-baked) of using too much cheese; so much that steam had trouble escaping from the crust. The top was overdone, while the crust, particularly just under the cheese, was wet, uncooked and doughy with overtones of slime.
